9 EXAMPLES OF CONVOLUTIONS OF MEASURES AND FUNCTIONS 285 PROBLEMS 1. Let G be a locally compact group and p, a positive nonzero bounded measure on G, such that jit * JJL = p.. (a) Show that S = Supp(^) is compact. (If /e 3f+(G) is not identically zero, remark that fji */must be constant on S, and use (14.9.2(iii)).) (b) Show that S is a compact subgroup of G and that p, is the Haar measure on S for which /x(S) = 1. (Use (a), Problem 2 of Section 12.9 and Problem 5 of Section 14.7.) 2. (a) Generalize (14.9.2(i)) to the case where 1 < p < + oo, by using Holder's inequal- ity (Section 13.11, Problem 12). (b) Let ju- be a bounded measure on G. Show that the norm of the continuous endo- morphism of LX(G, ]8) induced by /H-»/A */ (14.9.2) is equal to \\fi\\. (Let (/„) be a sequence satisfying the conditions of (14.11.1). If the norm in question were \\p\\ -a with a > 0, we should have N^/x, */„ * g) £ (|W - oWdfn * ff) ^ (HHI - a)Nito). Deduce that Nao(ju, */„) < ||jLt|| —a, and obtain a contradiction by letting n tend to +00.) (c) Under the hypotheses of (b), show that the norm of the continuous endomorphism of L°°(G, j3) induced by /*-*/* */ is equal to \\p.\\. (Reduce to the case where p. has compact support and has a continuous density with respect to |JK,|.) (d) Suppose that G is compact and JJL is positive. Show that, for 1 <p < -f oo, the norm of the continuous endomorphism of Lp(G,/2) induced by f\—»/A*/ is equal to HI. (e) Let G be a cyclic group of order 3. Give an example of a measure p, on G such that the norm of the endomorphism of L2(G, p) induced by f\-+p */is strictly less than ||/* ||. 3. (a) Let p, be a bounded measure on G. Show that we have NI(JLI */) = NA(/) for all / e JS?l(G, ]3) if and only if //, is of the form c - ex with c| = 1. (Using Problem 2(b), show that for each /e jf(G) we must have I f fdp = f \f\d\fji\. Deduce first that IJ J ^ = c|fi.|, where c is a constant such that |c| — 1, and then that yu is a point-measure.) (b) Take G to be a cyclic group of order 3. Give an example of a measure JJL on G which is not a point-measure and is such that N2(/x */) = N2(/) for all real-valued functions/on G. 4. Let G be a locally compact group and ft a left Haar measure on G. Let/be a bounded real-valued function on G, uniformly continuous with respect to a left-invariant distance on G. If /LA is any bounded measure on G, show that /x */(relative to ]8) is uniformly continuous with respect to a left-invariant distance on G. 5. Let G be a locally compact group and ]S a left Haar measure on G. (a) With the notation of Section 13.20, Problem 1, let (p,n) be a sequence of bounded measures on G which converges to /x with respect to the topology ^"3, and let (/„) be a sequence of functions in ^>1(G, j8) such that the sequence of bounded measures (/„ • j8) converges to / • ]8 with respect to the topology $~6. Show that the sequence of bounded measures (/An * (/„ • j8)) converges to (/u,B * (/ • j8)) with respect to ^6 (cf. Problem 4(d) of Section 14.7). (Use Problems 1 and 2 of Section 13.20.) (b) Let E5/2 denote the space of bounded real-valued functions on G which are uniformly continuous with respect to a left-invariant distance on G, and let ^5/2